MODIFICATION TO DARCY MODEL FOR HIGH PRESSURE AND HIGH

VELOCITY APPLICATIONS AND ASSOCIATED MIXED FINITE ELEMENT

FORMULATIONS

J. CHANG∗ AND K. B. NAKSHATRALA#

DEPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING, UNIVERSITY OF HOUSTON, HOUSTON 77204-4003.

Abstract. The Darcy model is based on a plethora of assumptions. One of the most important

assumptions is that the Darcy model assumes the drag coefficient to be constant. However, there

is irrefutable experimental evidence that viscosities of organic liquids and carbon-dioxide depend

on the pressure. Experiments have also shown that the drag varies nonlinearly with respect to

the velocity at high flow rates. In important technological applications like enhanced oil recovery

and geological carbon-dioxide sequestration, one encounters both high pressures and high flow

rates. It should be emphasized that flow characteristics and pressure variation under varying

drag are both quantitatively and qualitatively different from that of constant drag. Motivated

by experimental evidence, we consider the drag coefficient to depend on both the pressure and

velocity. We consider two major modifications to the Darcy model based on the Barus formula and

Forchheimer approximation. The proposed modifications to the Darcy model result in nonlinear

partial differential equations, which are not amenable to analytical solutions. To this end, we

present mixed finite element formulations based on least-squares (LS) formalism and variational

multi-scale (VMS) formalism for the resulting governing equations. The proposed modifications to

the Darcy model and its associated finite element formulations are used to solve realistic problems

with relevance to enhanced oil recovery. We also study the competition between the nonlinear

dependence of drag on the velocity and the dependence of viscosity on the pressure. To the best of

the authors’ knowledge such a systematic study has not been performed.

1. INTRODUCTION AND MOTIVATION

Understanding the flow of fluids through porous media plays a crucial role in various technological

applications (e.g., designing filters, enhanced oil recovery, geological carbon-dioxide sequestration)

and for mathematical modeling in various branches of engineering (e.g., civil engineering, petroleum

engineering, polymer engineering). Arguably, the most popular model in the studies on flow through

porous media is the Darcy model, which is named after the French hydraulics engineer Henry Darcy

who first proposed the equation in 1856 [1]. Darcy originally developed the model empirically based

Date: June 24, 2013.

Key words and phrases. Flow through porous media; Darcy equation; Forchheimer model; pressure-dependent

viscosity; ceiling flux; least-squares formalism; variational multi-scale formalism; enhanced oil recovery.

*Graduate student.

#Correspondence to: Dr. Kalyana Babu Nakshatrala, e-mail: knakshatrala@uh.edu, Phone:+1-713-743-4418.

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on the experiments on the flow of water in sand beds. However, the Darcy model can be given

firm mathematical basis at least in two different ways. One approach is by applying the volume

averaging theory on the Navier-Stokes equations [2]. The other approach is using the theory of

interacting continua (also known as mixture theory). For example, see reference [3, Introduction]).

In this paper, the latter approach will be employed.

1.1. Limitations of Darcy model, and its generalizations. Darcy equations model the flow

of an incompressible fluid in rigid porous media by stating that the (Darcy) velocity is linearly

proportional to the gradient of the pressure. It is important to note that the Darcy equation is

simply an approximation of the balance of linear momentum in the context of theory of interacting

continua. It merely predicts the flux but cannot predict stresses in solids. That is, this model

cannot be used with modification when there is deformation / damage of the porous solid (e.g., in

the case of hydraulic fracture). For completeness and future reference, let us enumerate the key

assumptions behind the Darcy model [3]. • There is no mass production of individual constituents

(i.e., there are no chemical reactions). • The porous solid is assumed to be rigid. Thus, the

balance laws for the solid are trivially satisfied. In particular, the stresses in the solid are what

they need to be to ensure that the balance of linear momentum is met. • The fluid is assumed to

be homogeneous and incompressible. • The velocity and its gradient are assumed to be small so

that the inertial effects can be ignored. • The partial stress in the fluid is that of an Euler fluid.

That is, there is no dissipation of energy between fluid layers. • The only interaction force is at the

fluid and pore boundaries. • Darcy model assumes that the drag coefficient is independent of the

pressure and the velocity. Experimental studies have shown that the viscosity of organic liquids

(and hence the drag coefficient) depend on the pressure [4,5] and the drag coefficient depends on the

velocity at high velocities [6, 7]. Therefore, Darcy model as it is not appropriate for applications

involving high pressure and high flow velocities. Application of Darcy model in such situations

can result in erroneous predictions of discharge fluxes and inaccurate pressure contours. While

several generalizations of the standard Darcy model have been proposed in the literature, none of

these studies addressed the study undertaken in this paper. In particular, the prior studies did not

address the combined effect of pressure-dependent viscosity and the dependence of drag coefficient

on the velocity.

1.1.1. Enhanced oil recovery. Over the years people have used Darcy model beyond its range of

applicability. One example of misuse is in the modeling enhanced oil recovery (EOR) applications.

As illustrated in Figure 1, steam / carbon-dioxide gas is injected into the ground through injection

wells. The gas create a pressure build up in the ground (i.e., the porous media) and pushes the

fluid (i.e., raw oil) out through the production wells. High pressures ranging from 10−100 MPa are2

Figure 1. A pictorial description of enhanced oil recovery.

[Source: https://www.llnl.gov/str/November01/Kirkendall.html]

employed, and such high pressures can lead to inaccurate flow estimates or pressure contours if the

original Darcy model is used. Oil reservoir simulations are tricky by nature because of the possibility

of having varying permeability within layers, impervious zones, non-rigid rock and soil formations,

and pockets of natural gases. Seismic imaging and field experimentation may not always return the

most accurate data so one must be extremely cautious when providing parameters to run numerical

models. Using the right Darcy modification(s) allows one to predict more accurate production rates,

help industries determine where to allocate their resources, and prevent environmental damage from

unintended cracking in the subsurface.

1.2. Mixed formulations. For the proposed model, we present mixed finite element formulations

based on two different approaches: least-squares (LS) finite element method [8] and variational

multi-scale (VMS) formalism [9]. It is well-known that care should be taken when working with

mixed formulations. In order to get stable results, a mixed formulation should either satisfy or

circumvent the Ladyzhenskaya-Babuska-Brezzi (LBB) stability condition [10]. Both the proposed

mixed formulations proposed in this paper circumvent the LBB condition.

The least-squares finite element method (LSFEM) is based on the minimization of the residuals in

a least-squares sense. One can always obtain a symmetric positive definite system of algebraic equa-

tions, even for non-self-adjoint problems. The LSFEM provides greater accuracy for the derivatives

of primal variables when compared to single-field formulations, boundary conditions are easy to

manage, the conformity of finite element spaces is sufficient to guarantee stability, and all variables

can use the same finite element space. For further discussion on the LSFEM, see references [8,11].

The VMS formalism adds stabilization terms to the classical mixed formulation. The stabilization

terms and the stabilization parameter can be derived in a consistent manner (e.g., see references3

[9, 12]). A mixed formulation based on the VMS formalism falls under the category of stabilized

methods, as in some sense the formulation is obtained by stabilizing the classical mixed formulation

[13]. Several studies as shown in references [14–18] have proposed various stabilized formulations

that provide accurate solutions of Darcy model through porous media, but none of these studies

considered the proposed model. Some notable mixed formulation based on the VMS formalism for

Darcy-type equations are references [3, 12, 19, 20]. We shall demonstrate in a subsequent section

that the proposed mixed formulation encompasses these mixed formulations.

1.2.1. Local mass balance. A common drawback of finite element formulations is that they need

not possess local / element-wise mass balance property. In particular, the mixed formulations

from both the LS and VMS formalisms do not possess the local mass balance property. It should

be noted that while it is possible to achieve local mass conservation for the LS formalism, one

would no longer be able to obtain continuous nodal quantities (e.g., see reference [21]). Several

independent studies [22–24] have successfully developed conservative finite element formulations

for flow through porous media problems, but none of them have been extended to modifications of

Darcy’s model. Another relevant work is reported in reference [25] in which the effect of error in

local mass balance on the transport of chemical species is studied. This study considered coupled

flow and transport problems, and the comparison is made between a VMS-based formulation and

the locally mass conservative Raviart-Thomas formulation. Herein, we shall perform a systematic

study on the performance of the proposed mixed formulations for various flow models with respect

to the local mass balance property. This study is intended to serve two purposes. First, it will

guide users of the finite element method (FEM) on the extent of the violation of local mass balance

under mixed formulations. Second, it will encourage researchers to improve the performance of

finite element formulations with respect to local balance under arbitrary interpolation functions for

the velocity and pressure.

1.3. Main contributions of this paper. Several contributions have been made in this paper with

respect to modeling of flow through porous media, associated mixed formulations, and numerical

solutions of representative problems. Some of the main ones are as follows:

(i) To propose a generalization of the Darcy model by taking into account both the dependence of

viscosity on the pressure and the dependence of drag coefficient on the velocity. The classical

Darcy-Forchheimer and the modified Darcy model that is considered in reference [3] will be

special cases of the generalized model considered in this paper. The generalization is referred

to as the modified Darcy-Forchheimer model.

(ii) To present some theoretical results pertaining to the modified Darcy-Forchheimer model, and

demonstrate their utility in validating numerical implementations.4

(iii) To develop a mixed formulation based on LS formalism for the modified Darcy-Forchheimer

model and study the effect of weighting on the convergence and accuracy of the solutions.

(iv) To construct a mixed formulation based on the VMS formalism for the modified Darcy-

Forchheimer model, which encompasses as special cases some of the existing mixed formula-

tions proposed for simpler models.

(v) To compare the numerical performances of VMS and LS based mixed formulations.

(vi) To document the local mass balance error under both these formalisms for the standard Darcy

model and for its generalizations.

(vii) It has been claimed in references [12, 26] that VMS formulation is the only known mixed

formulation that satisfies three-dimensional constant patch test under non-constant Jacobian

finite elements for the standard Darcy model. Herein, we show that the proposed LS based

formulation (which is different from the variational formulation proposed in these references)

also satisfies three-dimensional constant flow patch tests.

(viii) To discuss the implications and applicability of these modified models in numerical simulations

of enhanced oil recovery. It will also be show that the pressure profiles of Darcy-Forchheimer

are qualitatively and quantitatively different from that of a modification of Darcy model that

takes into account the dependence of viscosity on the pressure.

(ix) To illustrate an important competing effect due to the dependence of viscosity on the pressure

and the dependence of drag coefficient on the velocity. Specifically, to show the dependence of

drag coefficient on the velocity will create steep gradients in the pressure near the projection

well. On the other hand, the dependence of viscosity on the pressure creates steep pressure

gradients near the injection wells. However, both these effects give rise to ceiling flux.

1.4. Organization of the paper. The remainder of the paper is organized as follows. In Section

2 modifications to Darcy model using Barus formula and Forchheimer terms are presented. In

Section 3, mixed finite element formulations based on LS and VMS formalisms are proposed. In

Section 4, several representative test problems are solved to show the performance and convergence

of the proposed mixed finite element formulations, and to illustrate the predictive capabilities of the

modified Darcy-Forchheimer model. In Section 5, several representative problems with relevance

to enhanced oil recovery are simulated, and the numerical solutions from the various models and

formalisms are compared. Conclusions are drawn in Section 6.

2. GOVERNING EQUATIONS: DARCY MODEL AND ITS GENERALIZATION

Let Ω ⊂ Rnd be an open and bounded set, where “nd” denotes the number of spatial dimensions.

Let ∂Ω := cl(Ω) − Ω be the boundary (where cl(Ω) is the set closure of Ω), which is assumed

to be piecewise smooth. A spatial point in cl(Ω) is denoted by x. The gradient and divergence5

operators with respect to x are, respectively, denoted by grad[·] and div[·]. Let v : Ω→ Rnd denote

the “Darcy” velocity vector field (which is a homogenized velocity), and let p : Ω → R denote the

pressure field. The boundary is divided into two parts, denoted by Γv and Γp, such that Γv∩Γp = ∅and Γv ∪ Γp = ∂Ω. Γv is the part of the boundary on which the normal component of the velocity

is prescribed, and Γp is part of the boundary on which the pressure is prescribed.

We now consider the flow of an incompressible fluid through rigid porous media based on modi-

fications to the standard Darcy model. The governing equations take the following form:

α(v, p,x)v(x) + grad[p(x)] = ρb(x) in Ω (1a)

div[v(x)] = 0 in Ω (1b)

v(x) · n(x) = vn(x) on Γv (1c)

p(x) = p0(x) on Γp (1d)

where α is the drag coefficient (which can depend on the velocity and pressure, and can spatially

vary), vn(x) is the prescribed normal component of the velocity, p0(x) is the prescribed pressure, ρ

is the density of the fluid, b(x) is the specific body force, and n(x) is the unit outward normal vector

to the boundary. It can be shown that equation (1a) is an approximation to the balance of linear

momentum under the mathematical framework offered by the theory of interacting continua (e.g.,

see reference [3, Introduction]). A more thorough discussion on the theory of interacting continua

can be found in the several appendices of reference [27], Atkin and Craine [28], and Bowen [29].

2.1. Boundary conditions and well-posedness. We now briefly discuss the well-posedness of

the aforementioned boundary value problem given by equations (1a)–(1d) in the sense of Hadamard

[30]. If Γv = ∂Ω (i.e., the normal component of the velocity is prescribed on the entire boundary),

one has to meet the following compatibility condition for well-posedness:∫Γv=∂Ω

vn(x) dΓ = 0 (2)

which is a direct consequence of the divergence theorem. To wit,

0 =

∫Ω

div[v(x)] dΩ =

∫∂Ω

v(x) · n(x) dΓ =

∫∂Ωvn(x) dΓ (3)

Moreover, if Γp = ∅ (i.e., ∂Ω = Γv), one needs to augment the above equations (1a)–(1d) with an ad-

ditional condition for uniqueness of the solution. Otherwise, one cannot find the pressure uniquely.

In the Mathematics literature, the uniqueness is typically achieved by meeting the condition∫Ωp(x) dΩ = 0 (4)

which basically fixes the datum for the pressure. However, this approach is seldom used in a

computational setting as it is difficult to enforce the above condition numerically. An alternative6

is to fix the datum for the pressure by prescribing the pressure at a point, which is commonly

employed in various computational settings and is also employed in this paper.

It should also be noted that the no-slip boundary condition is not compatible with the Darcy

model and the generalization that is considered in this paper. A simple mathematical explanation

can be provided by noting that the inclusion of no-slip boundary condition (in addition to the

no-penetration boundary condition) will make the boundary value problem over-determined. Also,

it is noteworthy that the governing equations based on Darcy model are first-order (in terms of

number of derivatives) with respect to the field variables v(x) and p(x).

2.2. Darcy model, experimental evidence, and its generalization. The Darcy model as-

sumes that the drag coefficient is independent of the pressure and velocity. In addition, the Darcy

model assumes the drag coefficient to be of the form

α =µ

k(5)

where µ is the coefficient of viscosity of the fluid, and k is the coefficient of permeability. From

the above discussion it is evident that Darcy model cannot be employed for situations in which

the viscosity depends on the pressure, permeability depends on the (pore) pressure, or drag does

not depend linearly on the velocity of the fluid (i.e., the drag coefficient depends on the velocity).

Several experiments have shown unequivocally that these three situations occur in nature, which

will now be discussed.

2.2.1. Pressure-dependent viscosity. Bridgman [4] has shown that the viscosity of several organic

liquids depend on the pressure, and in fact, the dependence is exponential. Notable scientists such

as Andrade [31] and Barus [5] have performed laboratory experiments on liquids to determine the

relationship between pressure and viscosity. In recent years, research such as that in [32] has been

able to obtain empirical evidence to delineate and confirm the dependency of viscosity on pressure.

Furthermore, numerical studies have been performed in references [33,34] to record the differences

these pressure dependent viscosity equations make for several fluid problems like the Navier-Stokes

equation.

There are several ways one can generalized the standard Darcy model. For example, one can

model the friction between the layers of the fluid, which the standard Darcy model neglects. This is

approach taken by Brinkman (see references [35,36]). The research conducted in this paper focuses

on generalizing the standard Darcy model by modifying the drag to depend on the velocity and

the pressure.7

To account for the dependence of the viscosity (and hence the drag) on the pressure, Barus’

formula [37]will be used. The drag coefficient based on Barus’ formula can be rewritten as

α(p,x) =µ(p)

k(x)=

µ0

k(x)exp[βBp] (6)

where µ0 is the fixed viscosity of the fluid and βB is the Barus coefficient that is obtained experi-

mentally. This proposed modification states that the viscosity varies exponentially with pressure,

and one can determine the Barus coefficient βB using laboratory experiments, and its value for

common organic liquids (e.g., Naphthenic mineral oil) has been documented in the literature. For

example, see references [4, 38–40]).

2.2.2. High velocity flows and inertial effects. It has been experimentally observed that for high

velocity flows in porous media, the flux (and hence the flow rate) is not linearly proportional

to the gradient of the pressure. This can be explained by noting that inertial effects can play a

dominant role for high velocity flows. The standard Darcy model completely ignores inertial effects.

To address the nonlinear dependence of the flux on the gradient of the pressure for high velocity

flows, Philipp Forchheimer, an Austrian scientist (1852–1933), proposed that the drag coefficient

to depend on the velocity of the fluid [6]. Herein, the model that is obtained after incorporating

Forchheimer’s modification will be referred to as the Darcy-Forchheimer model.

It is noteworthy that the Darcy-Forchheimer model can be obtained from the Navier-Stokes

equations using the volume averaging method [41]. In typical geotechnical and civil engineering ap-

plications, one encounters low velocities so the inertial effects can be disregarded, and the standard

Darcy model is adequate. However, in high pressure applications like enhanced oil recovery one may

often encounter high velocities so inertial effects must be accounted for. The Darcy-Forchheimer

model is written as

α(v,x) =µ0

k(x)+ βF‖v‖ (7)

where βF is the Forchheimer or inertial coefficient, and ‖ · ‖ is the 2-norm. That is,

‖v‖ =√

v(x) · v(x) (8)

Several people have proposed their own experimental, theoretical, or computational formulations

for the Forchheimer coefficient (see reference [42]). For instance, one way to express βF is

βF =cFρ√kI

(9)

where cF is a dimensionless form-drag constant and kI is the inertial permeability, both of which

can be obtained experimentally. Successful mixed finite element formulations have been performed

on the Darcy-Forchheimer model in references [43, 44], but they all use different variants of the

Forchheimer coefficient. For the purpose of this paper, βF shall remain as a user-defined parameter.8

Remark 1. The laws of (Newtonian) mechanics are Galilean invariant. Therefore, one need to

construct the constitutive relations to be Galilean invariant so as to be consistent with the laws of

mechanics. At first glance, it may look like the model (7) and equation (1a) are not Galilean invari-

ant, as the velocity and the 2-norm of the velocity are not invariant under Galilean transformations.

However, it should be note that the velocity v(x) in these cases is the relative velocity between the

velocity of the fluid and the velocity of the porous solid. In Darcy-type models, it is tacitly assumed

that the porous solid is rigid and does not undergo any motion, which is also the case in this paper.

Therefore, the velocity v(x) is equal to the velocity of the fluid. Noting that the velocity v(x) is

the relative velocity is important, as the relative velocity and the norm of the relative velocity are

Galilean invariant. Hence, the model (7) is invariant under Galilean transformations. This will be

the case even with the other models considered in this paper.

Remark 2. Some porous solids exhibit strong correlation between permeability and porosity, and

studies presented in reference [45] show that the porosity is affected by the (pore) pressure. For these

porous solids, one can conclude that the pressure affects the permeability, which in turn will give

rise to the dependence of drag coefficient on the pressure. In this paper we do not solve any problem

that involves permeability depending on the pressure. However, the proposed mixed formulations

can be easily extended to handle such problems.

2.2.3. Proposed model: Modified Darcy-Forchheimer model. A major focus of this research will

study the effects of incorporating pressure-dependent viscosity into the Darcy-Forchheimer model.

The drag coefficient can then be rewritten as

α(v, p,x) =µ(p)

k(x)+ βF‖v‖ =

µ0

k(x)exp[βBp] + βF‖v‖ (10)

The proposed model is suitable for applications like enhanced oil recovery, geological carbon-dioxide

sequestration, and filtration process. The terms µ(p) and βF ‖v‖ (which are both nonlinear) can

have competitive effects, and neglecting either of these can give erroneous results for these appli-

cations.

It will now be shown that the modified Darcy-Forchheimer model is dissipative. That is, the

proposed constitutive model satisfies the second law of thermodynamics. Within the context of

theory of interacting continua for bodies undergoing isothermal processes [29], the total rate of

dissipation density at a spatial point x ∈ Ω, ξtotal(x), is written as

ξtotal(x) = ξsolid(x) + ξfluid(x) + ξinteraction(x) (11)

where ξsolid(x) and ξfluid(x) are the bulk rate of dissipation densities within the solid and the fluid,

and ξinteraction(x) is the bulk rate of dissipation density due to interaction of the solid and the fluid9

at their corresponding interfaces. Since the solid is assumed to be rigid,

ξsolid(x) = 0 (12)

The fluid is assumed to be perfect (i.e., an Euler fluid), so there is no (internal) dissipation within

the fluid. Thus we have

ξfluid(x) = 0 (13)

However, it should be emphasized that there is dissipation at the interface between the solid and

fluid, which is due to the drag. Hence the total rate of dissipation density at a spatial point x is

given by

ξtotal(x) = ξinteraction(x) = α(v, p,x) ‖v(x)‖2 (14)

where ‖ ·‖ is the 2-norm norm and v(x) is the relative velocity of the fluid with respect to the solid.

By ensuring that α(v, p,x) > 0 one can satisfy the second law of thermodynamics a priori. For

the modified Darcy-Forchheimer model given by equation (10) α(v, p,x) > 0, as µ0 > 0, k(x) > 0,

βF ≥ 0, ‖v‖ ≥ 0 and exp[·] > 0. The total dissipation due to drag in the entire domain takes the

following form:

Φ :=

∫Ωξinteraction(x) dΩ =

∫Ωα(v, p,x)v(x) · v(x) dΩ (15)

which is clearly non-negative.

Remark 3. A remark is warranted on the interpretation(s) of the quantity p(x), which was referred

to as the pressure earlier. Within the theory of constraints [46, 47], the quantity p(x) is the unde-

termined multiplier that arises due to the incompressibility constraint, which is given by equation

(1b). Note that p(x) is not referred to as a Lagrange multiplier as there are no Lagrange multipliers

under the mathematical framework for constraints that is outlined in references [46,47]. Under the

theory of interacting continua, the partial (Cauchy) stress in the fluid for Darcy model takes the

form

T(f) = −p(x)I, (16)

where I is the second-order identity tensor. Therefore, under the theory of interacting continua

framework, p(x) can be considered as the mechanical pressure in the fluid. Note that the mechanical

pressure is defined as the negative of the mean normal stress (see Batchelor [48]). Therefore,

for the modified Darcy-Forchheimer model, p(x) is both the mechanical pressure in the fluid, and

the undetermined multiplier to enforce the incompressibility constraint. The above discussion on

the precise identity and role of p(x) will be extremely important if one wants to make further10

generalizations / modifications to the proposed model. In particular, to extend the proposed model

to incorporate degradation and fracture of the porous solid, which will be part of our future work.

2.3. Some theoretical results for the modified Darcy-Forchheimer model. For the entire

discussion in this subsection, assume that Γv = ∂Ω. We shall also assume that the body force is a

conservative vector field. That is, there exists a scalar potential field φ(x) such that ρb(x) = grad[φ].

We shall refer to a vector field v(x) : Ω→ Rnd as kinematically admissible if it satisfies the following

conditions:

(i) v(x) is solenoidal (i.e., div[v(x)] = 0 in Ω)

(ii) v(x) satisfies the boundary conditions (i.e., v(x) · n(x) = vn(x) on ∂Ω)

Note that v(x) need not satisfy the balance of linear momentum given by equation (1a). We

now present an important property that the solutions of modified Darcy-Forchheimer equations

(1a)–(1d) satisfy: the minimum dissipation inequality.

Proposition 4. [Minimum dissipation inequality] Let v(x), p(x) be the solution of equations

(1a)–(1d). Any kinematically admissible vector field v(x) has to satisfy the following inequality:∫Ωα(v(x), p(x),x)v(x) · v(x) dΩ ≤

∫Ωα(v(x), p(x),x)v(x) · v(x) dΩ (17)

Proof. Let

δv(x) := v(x)− v(x) (18)

From the hypothesis, δv(x) satisfies the following equations:

δv(x) · n(x) = 0 ∀x ∈ ∂Ω (19a)

div[δv] = 0 ∀x ∈ Ω (19b)

Let us simplify the expression for the difference in the total dissipation due to drag:

δΦ :=

∫Ωα(v(x), p(x),x)v(x) · v(x) dΩ−

∫Ωα(v(x), p(x),x)v(x) · v(x) dΩ (20)

=

∫Ωα(v(x), p(x),x)δv(x) · (δv(x) + 2v(x)) dΩ (21)

=

∫Ωα(v(x), p(x),x)δv(x) · δv(x) dΩ + 2

∫Ωα(v(x), p(x),x)δv(x) · v(x) dΩ (22)

≥ 2

∫Ωα(v(x), p(x),x)δv(x) · v(x) dΩ = 2

∫Ωδv(x) · grad[φ(x)− p(x)] dΩ (23)

Using Green’s identity, we obtain the following inequality:

δΦ ≥ 2

∫∂Ωδv(x) · n(x) (φ(x)− p(x)) dΓ− 2

∫Ω

div[δv(x)] (φ(x)− p(x)) dΩ = 0 (24)

This completes the proof. 11

It is easy to obtain the following corollary for the case of constant drag coefficient.

Corollary 5. Let v1(x) and v2(x) be two Darcy velocities (i.e., they satisfy equations (1a)–(1d))

corresponding to two different constant drag coefficients but for the same conservative body force

and velocity boundary conditions, and for a given domain. Then the velocities satisfy∫Ω

v1(x) · v1(x) dΩ =

∫Ω

v2(x) · v2(x) dΩ (25)

Remark 6. Let the drag coefficient be independent of the velocity and the pressure. Let v1(x), p1(x)and v2(x), p2(x) be the solutions of equations (1a)–(1d) for the prescribed data b1(x), vn1(x)and b2(x), vn2(x), respectively. These fields satisfy the following relation:∫

Ωρb1(x) · v2(x) dΩ−

∫∂Ωp1(x)vn2(x) dΓ =

∫Ωρb2(x) · v1(x) dΩ−

∫∂Ωp2(x)vn1(x) dΓ (26)

The solutions to Darcy equations satisfy an identity similar to the Betti reciprocal relations in the

theory of elasticity [49,50] and creeping flows [51]. However, equation (26) is not valid for modified

Darcy-Forchheimer model.

The above results not only have theoretical significance but can also be invaluable in testing a

numerical implementation.

3. MIXED TWO-FIELD WEAK FORMULATIONS

It is, in general, not possible to obtain analytical solutions for the mathematical models presented

in the previous section. Hence, one may have to resort to numerical solutions. One of the main goals

of this paper is to present mixed finite element formulations based on the least-squares (LS) and

the variational multi-scale (VMS) formalisms for solving the boundary value problem arising from

the modified Darcy-Forchheimer model. Note that the standard Darcy, Forchheimer and modified

Darcy models are special cases of the proposed modified Darcy-Forchheimer model. Therefore, the

proposed mixed formulations can be used to solve these models, and encompass some of the prior

mixed formulations that have developed for these simpler models.

The following function spaces will be used in the remainder of this paper:

P :=p(x) ∈ H1(Ω) | p(x) = p0(x) on Γp

(27a)

Q :=q(x) ∈ H1(Ω) | q(x) = 0 on Γp

(27b)

Q :=q(x) ∈ H1(Ω)

, (27c)

V :=

v(x) ∈ (L2(Ω))nd | div[v] ∈ L2(Ω), v(x) · n(x) = vn(x) on Γv

(27d)

W :=

w(x) ∈ (L2(Ω))nd | div[w] ∈ L2(Ω), w(x) · n(x) = 0 on Γv

(27e)

12

where L2(Ω) and H1(Ω) are standard Sobolev spaces [10]. Note that two different function spaces

are defined for the pressure trial function. If the pressure is prescribed strongly on Γp then the

function space given in equation (27a) will be used for the pressure trial function, and the function

space given in equation (27b) will be used for the pressure test function. If the pressure is prescribed

weakly on Γp then the function space given in equation (27c) will be used for both trial and test

functions of the pressure. It should be emphasized that both L2(Ω) and H1(Ω) are Hilbert spaces

under the standard L2 inner-product [52]. The standard L2 inner-product over a set K will be

denoted as (·; ·)K , and is defined as

(a; b)K :=

∫Ka · b dK (28)

For simplicity, the subscript K will be dropped if K = Ω. Note that for volume integrals K ⊆ Ω

and for surface integrals K ⊆ ∂Ω. In a subsequent section on numerical results, the error will be

measured in L2 norm and H1 seminorm. To this end, the L2 norm on Ω is defined as

‖a‖L2(Ω) :=

√∫Ωa · a dΩ (29)

The H1 seminorm on Ω is defined as

|a|H1(Ω) :=

√∫Ω

grad[a] · grad[a] dΩ (30)

The H1 norm on Ω can then be defined as

‖a‖H1(Ω) :=√‖a‖2L2(Ω) + |a|2

H1(Ω)(31)

For further details on inner-product spaces and normed spaces, see references [53,54].

The aforementioned modifications to the standard Darcy model result in nonlinear partial dif-

ferential equations, as the drag coefficient depends on the pressure and/or the velocity. To solve

the resulting nonlinear equations, linearization is first performed, and then the LS and VMS for-

malisms shall be utilized to construct mixed two-field weak formulations. To this end, let us define

the following linearization functionals:

D(i+1) := ϑ

(∂α

∂pv(i)

)p(i+1) + ϑ

(∂α

∂v⊗ v(i)

)v(i+1) (32)

D(i) := ϑ

(∂α

∂pv(i)

)p(i) + ϑ

(∂α

∂v⊗ v(i)

)v(i) (33)

G := ϑ

(∂α

∂pv(i)

)q + ϑ

(∂α

∂v⊗ v(i)

)w (34)

where superscripts (i) and (i+1) represent solutions for the current and next iteration respectively,

⊗ denotes the standard tensor product [55], and ϑ ∈ [0, 1] is a user-defined parameter to choose13

the type of linearization. One can achieve Picard’s linearization by choosing ϑ = 0 and consistent

linearization by choosing ϑ = 1.

Remark 7. It should be noted that v, p, w, q, µ, b, k, and n are all functions of x. The drag

coefficient and its derivatives will be functions of p(i), v(i) and x. For notational simplicity, these

dependencies will not be explicitly indicated.

3.1. A mixed formulation based on least-squares formalism. Consider an abstract mathe-

matical problem defined by a set of partial differential equations in the form:

Lu = f in Ω (35)

Bu = 0 in Γ (36)

where L is the differential operator, B is the boundary operator, u is the unknown vector, and f is

the forcing vector. A corresponding least-squares functional can be constructed as follows:

Π[u] =1

2

∫Ω‖Lu− f‖2 dΩ +

1

2

∫Γ‖Bu− 0‖2 dΓ (37)

A weak form based on least-squares formalism can be obtained by requiring the Gateaux variation

to vanish along any w that satisfies the essential boundary conditions. That is,

δΠ[w,u] = 0 ∀w (38)

where

δΠ[w,u] := limε→0

Π[w + εu]−Π[w]

ε≡[d

dεΠ[w + εu]

]ε=0

(39)

provided the limit exists. For further details on the Gateaux variation see references [56–58].

Studies in reference [59] have shown that minimizing the problem after linearization produces

more accurate results. Also, minimizing a least-squares-based functional before linearization will

create additional terms in the resulting weak formulation and significantly increase the difficulty

of implementation, so we shall employ the former approach. Inserting equations (32) and (33) into

equation (1a) results in the following governing equations:

αv(i+1) +D(i+1) −D(i) + grad[p(i+1)] = ρb in Ω (40a)

div[v(i+1)] = 0 in Ω (40b)

v(i+1) · n = vn on Γv (40c)

p(i+1) = p0 on Γp (40d)

In reference [60], it has been shown that for the Navier-Stokes equation, an introduction of a mesh

dependent variable in the LS formulation greatly improves the accuracy of the solution. Thus for14

the Darcy modifications, two variants of the LS formulation will be considered by employing the

following weights:

A =

I weight 1

αI weight 2(41)

For all the models considered in this paper, the second-order tensor A is symmetric and positive

definite. This implies that the tensor is invertible. In addition, the square root theorem ensures

that its square root exists [61]. Employing the minimization approach on equations (40a) and (40b)

results in the functional

ΠLS[v(i+1), p(i+1)] :=1

2

∫Ω

∥∥∥A−1/2(αv(i+1) +D(i+1) −D(i) + grad[p(i+1)]− ρb)∥∥∥2

dΩ

+1

2

∫Ω

∥∥∥div[v(i+1)]∥∥∥2

dΩ (42)

Let v(i+1) → v(i+1) + εw and p(i+1) → p(i+1) + εq where v(i+1) and w are the velocity trial and

test functions respectively and p(i+1) and q are the pressure trial and test functions respectively.

Applying the Gateaux variation on equation (42) results in the functional

δΠLS[v(i+1), p(i+1); w, q] =

[d

dεΠLS[v(i+1) + εw, p(i+1) + εq]

]ε=0

=

∫Ω

(αw + G + grad[q]) ·A−1(αv(i+1) +D(i+1) −D(i) + grad[p(i+1)]− ρb

)+ div[w] · div[v(i+1)]dΩ (43)

and setting it equal to zero gives the weak form. The final statement for the modified Darcy-

Forchheimer model can be rearranged and written as follows: Given v(i) andp(i) find v(i+1) ∈ Vand p(i+1) ∈ P such that we have(

αw; A−1αv(i+1))

+(αw; A−1D(i+1)

)+(αw; A−1grad[p(i+1)]

)+(G; A−1αv(i+1)

)+(G; A−1D(i+1)

)+(G; A−1grad[p(i+1)]

)+(

grad[q]; A−1αv(i+1))

+(

grad[q]; A−1D(i+1))

+(

grad[q]; A−1grad[p(i+1)])

+(

div[w]; div[v(i+1)])

=(αw; A−1ρb

)+(G; A−1ρb

)+(grad[q]; A−1ρb

)+(αw; A−1D(i)

)+(G; A−1D(i)

)+(

grad[q]; A−1D(i))

∀w ∈ W, ∀q ∈ Q (44)

3.2. A mixed formulation based on the variational multi-scale formalism. Following the

derivation given in reference [12], one can derive a mixed formulation based on VMS formalism.

It should be noted that in the previous derivations, the governing equations were not linearized

and were solved using a Newton-Raphson approach. It should also be noted that the pressure15

boundary condition is weakly prescribed (i.e., a Neumann boundary condition) and acts normal to

the boundary so the function space in equation (27c) is utilized.

After incorporating linearization terms into the governing equations, the resulting weak form

based on the VMS formalism can be written as follows: Given v(i) and p(i) find v(i+1) ∈ V and

p(i+1) ∈ P such that we have(w;αv(i+1)

)+(w;D(i+1)

)−(

div[w]; p(i+1))

+ (w · n; p0)Γp −(q; div[v(i+1)]

)−1

2

(αw + grad[q];α−1

(αv(i+1) +D(i+1) + grad[p(i+1)]

))︸ ︷︷ ︸

stabilization term

=(w; ρb +D(i)

)−1

2

(αw + grad[q];α−1

(ρb +D(i)

))︸ ︷︷ ︸

stabilization term

∀w ∈ W, ∀q ∈ Q (45)

The proposed VMS formulation encompasses some of the existing mixed formulations proposed

for simpler models. For the standard Darcy model, this weak formulation reduces to the one

presented in reference [12]. For the modified Darcy model, this formulation with ϑ = 0 (i.e.,

Picard linearization) reduces to the one presented in reference [20]. Reference [3] considered the

modified Darcy model. However, the mixed formulation proposed in reference [3] took a different

approach by first constructing a weak formulation based on VMS formalism before linearization.

The resulting nonlinear equations are then solved using the Newton-Raphson method. Algorithm

1 outlines the steps in implementing the proposed mixed finite element formulations.

4. NUMERICAL BENCHMARK TESTS

4.1. Dimensionless form of equations. Numerical studies for subsurface flows like enhanced

oil recovery can be displayed in dimensionless form, thus allowing scaling to real flow conditions.

The governing equations are non-dimensionalized by choosing primary variables that seem ap-

propriate. This non-dimensional procedure is different from the standard non-dimensionalization

procedure for incompressible Navier-Stokes in the choice of primary variables (in the standard non-

dimensionalization of Navier-Stokes equations, one employs characteristic velocity V , characteristic

length L and density of the fluid ρ as primary variables). Also, the present non-dimensionalization

is different and seems more appropriate than the one employed in reference [3] for the chosen

applications.

All non-dimensional quantities are denoted using a superposed bar. Let L (reference length in

the problem), g (acceleration due to gravity) and patm (atmospheric pressure) be the reference16

Algorithm 1 Pseudocode for the nonlinear FEA.

Set (i) = 1;

Initialize data v(i) = 1 and p(i) = 1

while true do . nonlinear solver

if (i) > maximum number of iterations then

break . solution did not converge

end if

Get α using v(i) and p(i)

Assemble stiffness matrices and forcing vectors

Solve and obtain v(i+1) and p(i+1)

if∥∥v(i+1) − v(i)

∥∥ and∥∥p(i+1) − p(i)

∥∥ < εTOL then

break . solution has converged

else

v(i) ← v(i+1) and p(i) ← p(i+1)

(i)← (i+ 1)

end if

end while

quantities. The following non-dimensional quantities are then defined:

x =x

L, v =

v√gL

, b =b

g, p =

p

patm, ρ =

ρgL

patm, k =

k

L2

βB = βBpatm, βF =βFgL

2

patm, α = α

√gL3

patm, µ0 =

µ0

√g/L

patm(46)

The scaled domain Ωscaled is defined as follows: a point in space with position vector x ∈ Ωscaled

corresponds to the same point with position vector given by x = xL ∈ Ω. Similarly, one can define

the scaled boundaries for Γvscaled and Γpscaled. Using the above non-dimensionalization procedure,

the governing equations (1a)–(1d) can be written as follows:

α(v, p, x)v + grad[p(x)] = ρ b(x) in Ωscaled (47a)

div[v(x)] = 0 in Ωscaled (47b)

v(x) · n(x) = v0(x) on Γvscaled (47c)

p(x) = p0(x) on Γtscaled (47d)

4.2. Numerical h-convergence. A finite element formulation is said to be convergent if the

numerical solutions tend to the exact solution with mesh refinement. This section will perform an17

Figure 2. Typical structured meshes using quadrilateral (left) and triangular

(right) finite elements, which are employed in the h-numerical convergence stud-

ies.

h-convergence analysis on all Darcy models where h is taken to be the edge length for quadrilateral

elements and the short-edge length for triangular elements.

Consider a unit square as the computational domain. For this and all subsequent numerical

studies, the FEM utilizes structured meshes as depicted in Figure 2. The velocity and pressure

functions for this problem are:

v(x, y) =

2y(x+ y),

4x− y2.

(48a)

p(x, y) = 10− xy − sin(πx)sin(πy) (48b)

Inserting the velocity and pressure functions back into the Darcy equation results in the following

specific body force function:

b(x, y) =1

ρ

α2y(x+ y)− πcos(πx)sin(πy)− yα(4x− y2)− πcos(πy)sin(πx)− x

(49)

Normal components of the velocity function in equation (48) are prescribed as the boundary

Table 1. User-defined inputs for the numerical h-convergence problem.

Parameters Value

βB 0.1

βF 0.5

α 1

ρ 1

ϑ 1

h-sizes 1/4, 1/8, 1/16, 1/32, 1/64

18

(a) x-velocity (b) y-velocity

(c) velocity field (d) pressure contour

Figure 3. Numerical h-convergence: contours of analytical solution.

condition. A pressure of 10 is also prescribed at the bottom left corner to ensure uniqueness in

the solution. Using the parameters listed in Table 1, Figure 3 depicts the analytical velocity and

pressure solutions to which the finite element solutions shall be compared with. Since neither the

pressure nor velocity functions depend on the drag coefficient, only the specific body force varies

with respect to each Darcy model. The four Darcy models used are: the original Darcy (D), modified

Barus (MB), Darcy-Forchheimer (F), and modified Darcy-Forchheimer Barus (MBF) models. The

L2 norm and H1 seminorm error slopes for the modified Darcy-Forchheimer Barus models are

depicted in Figure 4. Four-node quadrilateral (Q4) and three-node triangular (T3) elements are

used to solve the problems, and Table 2 lists all the error slopes for the rest of the models.

It can be seen that the numerical solutions perform well; converged solutions should have error

slopes of approximately -2.00 and -1.00 for L2 norm and H1 seminorm respectively. Though19

1 2 3 4 5−10

−8

−6

−4

−2

0

slope = −2.00

slope = −1.98

slope = −1.00

slope = −1.00

−log(h)

log(

erro

r)

velocity L2pressure L2velocity H1pressure H1

(a) LS: Q4 elements

1 2 3 4 5−10

−8

−6

−4

−2

0

slope = −1.94

slope = −1.64

slope = −1.00

slope = −0.95

−log(h)

log(

erro

r)

velocity L2pressure L2velocity H1pressure H1

(b) LS: T3 elements

1 2 3 4 5−10

−8

−6

−4

−2

0

slope = −2.00

slope = −2.02

slope = −1.02

slope = −1.00

−log(h)

log(

erro

r)

velocity L2pressure L2velocity H1pressure H1

(c) VMS: Q4 elements

1 2 3 4 5−8

−7

−6

−5

−4

−3

−2

−1

0

slope = −1.86

slope = −1.72

slope = −1.11

slope = −0.96

−log(h)

log(

erro

r)

velocity L2pressure L2velocity H1pressure H1

(d) VMS: T3 elements

Figure 4. Numerical h-convergence: error slopes for the MBF model

not shown, it has been found that the error slopes for all six models are similar to one another.

Quadrilateral elements tend to exhibit faster convergence rates than triangular elements, and one

can expect even faster rates for higher order elements like the nine-node quadrilateral Q9 and

six-node triangular T6.

4.3. Quarter five-spot problem. This section presents numerical results for a quarter spot prob-

lem as depicted in Figure 5. In many enhance oil recovery applications, there is an injection well

centered around four production wells. When carbon-dioxide is injected into the ground, the pres-

sure build up pushes oil out through the four injection wells. This schematic forms what is often

known as the five spot problem. Numerical results will exhibit elliptic singularities near the in-

jection and production wells and provide a good benchmark to test the robustness of the finite

element formulations. Due to the symmetric nature of the problem, only the top right quadrant is20

Table 2. Numerical h-convergence slopes for various Darcy models.

LS formalism VMS formalism

D MB F MBF D MB F MBF

Q4 L2 error v -2.00 -2.00 -1.99 -2.00 -1.99 -2.00 -1.99 -2.00

Q4 H1 error v -1.00 -1.00 -1.00 -1.00 -1.14 -1.04 -1.05 -1.02

Q4 L2 error p -1.95 -1.96 -1.99 -1.98 -2.01 -2.03 -2.02 -2.02

Q4 H1 error p -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00

T3 L2 error v - 1.97 -1.97 -1.84 -1.94 -1.84 -1.86 -1.81 -1.86

T3 H1 error v -1.01 -1.01 -1.00 -1.00 -1.13 -1.13 -1.07 -1.11

T3 L2 error p -1.62 -1.65 -1.62 -1.64 -1.69 -1.71 -1.70 -1.72

T3 H1 error p -0.95 -0.95 -0.95 -0.95 -0.96 -0.96 -0.96 -0.96

Injection well

Production well Production well

Production well Production well

vy = 0

vy = 0

vx=

0

vx=

0

L = 1

L=

1

x

y

Figure 5. Quarter five-spot problem: A pictorial description.

considered in the analysis. There is no specific body force or volumetric/sink source, and a pressure

of p0 = 1 is prescribed at the production well or top right node. Since it has been shown in previous

sections that the FEM developed performs well for both Q4 and T3 elements, only quadrilateral

elements Q4 and Q9 will be used to simulate all proceeding numerical simulations.

Consider a case where there is√

2 units of flow through the unit square quadrant. To attain

this flow rate, one needs to know the amount of pressure needed at the injection well (i.e., the

bottom left node). Using Q4 elements and the parameters listed in Table 3, Figure 6 depicts the

qualitative velocity vector field and the pressure contour. While both formalisms exhibit similar

pressure contours, the velocity vector field generated from the LS method exhibits poor dispersion21

(a) LS velocity vector field (b) LS pressure contour

(c) VMS velocity vector field (d) VMS pressure contour

Figure 6. Quarter five-spot problem: Q4 solutions for Darcy model

Table 3. User-defined inputs for the quarter five-spot problem: Darcy model.

Parameters Value

α 1

ρ 1

b 0

Nele 400

p(1, 1) 1

vx(0, 0), vy(0, 0) 1

vx(1, 1), vy(1, 1) 1

22

(a) LS velocity vector field (b) LS pressure contour

(c) VMS velocity vector field (d) VMS pressure contour

Figure 7. Quarter five-spot problem: Q9 solutions for Darcy model

of flow concentration at both wells. Intuitively, the profile of Figure 4.3 makes little to no physical

sense so when using the LSFEM, neither Q4 nor any other first order elements can be used to

accurately model velocity contours.

However, when higher order elements are used, the LS velocity vector field resembles that of the

VMS. Figure 7 depicts the results using Q9 elements. It should be noted that the pressure contours

remain the same regardless of the element order used. Realistic pressure profiles can be obtained

using either formalism or element type, but obtaining velocity and flow solutions with the LSFEM

necessitates the use of Q9 or higher ordered elements.

4.3.1. Least-squares weighting. For all original Darcy model problems up to this point, the non-

dimensionalized drag coefficient equals one, so the two possible LS weightings A in equation (41)

would be the same. When α no longer equals one, weighting number 2 begins to have a significant23

Table 4. Quarter five-spot problem: Injection pressure comparison for different LS weightings.

α : 1 20 50 100 250 500 1000

LS weight 1 Q4: 1.26 6.03 12.56 21.57 47.29 91.38 180.57

LS weight 1 Q9: 1.27 6.38 14.44 27.76 67.17 132.63 263.77

LS weight 2 Q4: 1.26 6.19 13.90 26.67 64.43 126.00 244.97

LS weight 2 Q9: 1.27 6.38 14.46 27.92 68.31 135.60 269.37

VMS Q4: 1.27 6.37 14.42 27.84 68.09 135.18 269.37

VMS Q9: 1.27 6.38 14.46 27.93 68.32 135.63 270.27

impact on the numerical solutions. Herein, the injection pressure shall be obtained using various

drag coefficients (all other user-defined parameters are as stated in Table 3). The VMS formalism

serves as a benchmark for the two LS weightings. From Table 4, it is seen that a divergence in the

solutions occurs as the drag coefficient increases. LS formalism using Q9 elements has comparable

stiffness to that of VMS formalism using Q4 elements, but as the drag increases, the VMS formalism

using Q9 elements requires larger and larger pressures. Nonetheless, all the solutions show a linear

relationship between drag and injection pressure. For highly viscous or lowly permeable reservoirs,

one has to apply more pressure in order to attain or expect a certain flow. If drag is a function of

pressure and/or velocity, one can expect even greater injection pressures.

4.3.2. Comparison of beta coefficients, pressure profiles, and linearization types. This next study

shall illustrate the effect the Barus and Forchheimer coefficients have on the pressure profile and

convergence of residuals. For pressure dependent viscosities, the Barus coefficient for most oils range

between 15 to 35 GPa−1 (see reference [62]) which translates to a non-dimensionalized coefficient of

roughly 0.001 to 0.004. However, for the purpose of this experiment, much higher Barus coefficients

shall be used. The same Barus coefficients used will also be used for the Forchheimer coefficients.

The relationship between the coefficients and the number of iterations needed to converge the

residuals will also be shown for both linearization types.

Figure 8 depicts the pressure profile diagonally across the quarter region. Various beta values

were used for the modified Barus and Darcy-Forchheimer models (assume both βB and βF to be

denoted by the same β). Overall the LS and VMS formalisms generate similar results. As the

coefficient β increase, the pressure gradients at the two wells steepen. The modified Barus model

exhibits the steepest gradients which is expected. The Darcy-Forchheimer solutions also exhibit

increases in the injection pressure, but the qualitative nature of the pressure gradients near the

wells are slightly different. Since the modified and Darcy-Forchheimer models rely on separate

non-Darcy coefficients and different dependent variables, no true comparisons can be drawn. In the24

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

x/y

pres

sure

beta = 0.10beta = 0.50beta = 0.75beta = 1.00

(a) Modified Barus - LS Q9

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

x/y

pres

sure

beta = 0.10beta = 0.50beta = 0.70beta = 1.00

(b) Modified Barus - VMS Q9

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

x/y

pres

sure

beta = 0.10beta = 0.50beta = 0.70beta = 1.00

(c) Darcy-Forchheimer - LS Q9

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

x/y

pres

sure

beta = 0.10beta = 0.50beta = 0.70beta = 1.00

(d) Darcy-Forchheimer - VMS Q9

Figure 8. Quarter five-spot problem: pressure profile vs various βB and βF

next Section however, distinction of results from pressure dependent and velocity dependent drag

coefficients will become more evident.

It should be noted that the pressure profiles in Figure 8 were generated using Picard’s lineariza-

tion (i.e. ϑ = 0). While Picard’s and consistent linearization theoretically yield the same results,

the residual convergence schemes differ. Table 5 contains the iteration count and residual norms

for the modified Barus model evaluated at βB = 0.6. Consistent linearization exhibits terminal

quadratic convergence whereas Picard’s linearization exhibits terminal linear convergence. As the

betas and/or applied pressure increases, the number of iterations needed increases.

4.3.3. Modified Darcy-Forchheimer numerical results. So far it has been established in this section

that quadratic elements and LS weight 2 are preferred for the LS formalism. Numerical simulations

have also shown that high Barus and Forchheimer coefficients yield results that differ quantitatively25

Table 5. Quarter five-spot problem: Picard vs. consistent linearization iteration

counts for modified Barus model with βB = 0.6. The top table corresponds with LS

formalism, and the bottom table corresponds with VMS formalism. Q9 elements

are used

LS formalism: Picard’s linearization consistent linearization

Iteration no. (i) v residual p residual v residual p residual

1 1.637285e+01 2.793981e-01 1.637323e+01 2.853694e-01

2 6.735667e-04 6.821094e-03 1.578203e-02 6.444641e-02

3 8.721274e-05 1.096770e-03 7.696835e-04 1.025242e-02

4 8.150045e-06 1.225905e-04 7.976516e-07 2.339612e-05

5 5.881603e-07 1.019954e-05 2.200057e-10 8.554086e-09

6 3.455235e-08 6.755041e-07 5.434345e-14 3.075479e-12

7 1.712434e-09 3.720906e-08

8 7.340079e-11 1.755375e-09

9 2.770668e-12 7.240625e-11

VMS formalism: Picard’s linearization consistent linearization

Iteration no. (i) v residual p residual v residual p residual

1 6.474444e-02 1.376963e-01 2.086226e-01 1.376963e-01

2 5.513505e-03 3.454112e-03 1.506223e-01 3.675604e-02

3 4.187925e-04 5.555865e-04 4.681837e-03 1.500411e-02

4 2.726750e-05 6.208117e-05 6.326269e-06 9.377515e-05

5 1.551030e-06 5.160684e-06 9.996752e-10 5.528937e-08

6 7.728631e-08 3.412260e-07 1.985333e-13 1.116834e-11

7 3.389569e-09 1.874543e-08

8 1.318682e-10 8.807571e-10

from the original Darcy model. This next example shall study the effects of combining the Darcy

models and employs a finer mesh.

Using the same boundary conditions as before, a non-dimensionalized Barus and Forchheimer

coefficient of 0.5 and 0.5 shall be used. Since the qualitative nature of the pressure contours are

similar no matter which model is used, only the required injection pressure is listed. It can be seen

from the results in Table 6 that refining the mesh lowers the pressure. For problems where flow

quantities are fixed, coarse meshes over predict the required pressure needed. The Barus, linear,

and Forchheimer models all predict pressures greater than that of the original Darcy model, and26

Table 6. Quarter five-spot problem: expected injection pressures for various Darcy

models and mesh sizes

Nele formalism D MB F MBF

400 LS 1.2692 1.5017 1.3382 1.5806

400 VMS 1.2693 1.5020 1.3382 1.5809

900 LS 1.1967 1.3538 1.2430 1.4045

900 VMS 1.1967 1.3539 1.2430 1.4047

4 5 6 7 8 9 10 11 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1/h

Dis

sipa

tion

DarcyModified BarusDarcy−ForchheimerModified Barus Forchheimer

(a) LS formalism

4 5 6 7 8 9 10 11 120

0.05

0.1

0.15

0.2

0.25

1/h

Dis

sipa

tion

DarcyModified BarusDarcy−ForchheimerModified Barus Forchheimer

(b) VMS formalism

Figure 9. Quarter five-spot problem: dissipation vs h-size

when the modified Darcy-Forchheimer models are employed, we get even higher pressures. The

original Darcy model under predicts the amount of pressure needed so it is important to use the

modified Darcy-Forchheimer models to attain an accurate visualization of the pressure contours.

4.3.4. Minimum dissipation. Dissipation shall now be measured for this problem using various mesh

sizes (Barus and Forchheimer coefficients of 0.1 and 0.5 shall be used respectively). The h-sizes27

used for this problem are 1/4, 1/6, 1/8, 1/10, and 1/12. All four Darcy models are evaluated using

both formalisms, and the results can be found in Figure 9. It is seen that as the mesh gets finer

the dissipation becomes smaller hence satisfying the minimum dissipation inequality. Though the

LS formalism yields slightly higher dissipations, the results of both formalisms are very similar.

4.4. Three-dimensional constant flow. It has been claimed in references [12,26] that the VMS

mixed formulation for Darcy model is the only known mixed formulation that satisfies constant

flow patch test in three dimensions on non-constant Jacobian finite elements. We show that the

mixed formulation based on least-squares formalism also satisfy the constant patch test in three

dimensions for Darcy model. We shall also use the test problem to show that the proposed mixed

formulations perform well even for other modifications of the Darcy model. (It should be emphasized

that this problem can be considered as a patch test only for Darcy model, and not for modified

Darcy-Forchheimer, as the exact solution under the modified Darcy-Forchheimer model will not be

neither linear nor constant.)

The computational domain is a unit cube, which is meshed using eight-node brick elements.

Normal components of the velocity are prescribed as unity on the y-z planes at x = 0 and x =

1. The other four planes have normal components of velocity equal to zero, and a pressure value

of zero is prescribed at (0,0,0) to ensure uniqueness to the solution. Using the values defined in

Table 7, the LS results for the original Darcy, modified Barus, Darcy-Forchheimer, and modified

Darcy-Forchheimer Barus models are shown in Figure 10. Clearly, the LS-based mixed formulation

satisfies the constant patch test.

5. ENHANCED OIL RECOVERY APPLICATIONS

It has been shown in the previous section that the proposed mixed formulations perform well

for the benchmark tests and that various modifications to the Darcy model have a significant

impact on the results. This section focuses on relevant enhanced oil recovery applications, which

Table 7. User-defined inputs for the three-dimensional problem

Parameter Value

βB 0.5

βF 1

k 1

µ0 1

ϑ 1

b(x) 0

Nele 216

28

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 10. Three dimensional problem: Pressure contours using LS formalism.

are more complex by nature. Pressure contours, flow rates, and errors in the local/element-wise

mass balance. Q9 elements shall be used for each problem.

5.1. Oil reservoir problem. For high pressure applications like enhanced oil recovery, one is

interested in the quantitative and qualitative nature of the pressure contours and velocities within

the oil reservoir. The pictorial description of a typical oil reservoir is depicted in Figure 11. Injection

wells are located on either side the production well, and carbon-dioxide is pumped into the reservoir

to ease the extraction of raw oil through the production well. The parameters used for this study are

listed in Table 8. All Darcy models and finite element formulations are expected to yield differing

flow patterns, but the general qualitative velocity vector can be depicted in Figure 12. As the oil

fluid nears the production well, the Darcy velocities increases. Pressure contours within the oil29

Table 8. User-defined inputs for the oil reservoir problem

Parameter Value

βB 0.005

βF 0.01

k 1

µ0 1

ρ 1

ϑ 1

b(x)

0;−1

Nele 1600

penh 1000

reservoirs are important to know because high pressures can result in cracking of the solid. Figures

13 and 14 contain the pressure contours using the LS and VMS formalisms respectively.

It can be seen from each model that the pressure contours within the reservoirs vary both qualita-

tively and quantitatively. For the Barus model, there are steep pressure gradients near the injection

well, and the pressures within the reservoir are generally smaller than that of the Darcy model.

However, the Darcy-Forchheimer models exhibits steep pressure gradients near the production well,

thus predicting higher pressures throughout the reservoir. While pressure dependent viscosity may

yield favorable pressure contours, one has to account for increases in pressure due to inertial effects,

so combining the Barus and Forchheimer models should yield the most accurate results. Figure 15

depicts the pressure profiles of all models and formalisms at the top most interface of the reservoir.

It should be noted that there are some minor differences in the pressure profiles between the LS

and VMS formalisms. While both formalisms have strongly prescribed velocity boundary condi-

tions, the VMS boundary condition for pressures are weakly prescribed and consequently exhibit

some oscillations. The oscillations diminish with mesh refinement, but one must recognize the

potential ramifications oscillatory boundary conditions may have on the solutions, especially for

more complex prescribed pressures.

In reservoir simulations, another quantity of interest is the outflow of raw oil. The flow rate or

total flux at the production well is calculated using∫Γp

v · n dΓ, (50)

where Γp corresponds with the prescribed atmospheric pressure boundary. In Figure 16, a com-

parison of flow rates versus prescribed pressures is shown for both formalisms. The original Darcy

models predict a linear relationship between prescribed pressures and flow rates but the non-linear30

Darcy models exhibit ceiling fluxes. As the pressure increases, the original Darcy models becomes

increasingly unreliable as it over predicts the amount of oil production one can expect. It is inter-

esting to note that for both the Darcy and Barus models, the LS formalism predicts higher flows

for a fixed injection pressure whereas the VMS formalism predicts higher Forchheimer flow rates.

Nevertheless, the ceiling fluxes for the Barus and Forchheimer models differ for various betas, but

combining the two models will always yield smaller flow rates.

As stated in Section 1, neither the LS nor VMS formalisms have local mass conservation. The

ratios of local mass balance errors over the total predicted flux for the modified Darcy-Forchheimer

Barus models are shown in Figure 17 and 18. When one encounters high velocity contours, one can

also expect higher local mass balancing errors. The calculations show that all models exhibit the

greatest errors near the production wells. It is interesting to note that while both formalisms predict

roughly the same velocity flow rates, the VMS formalism shows greater local mass balancing error.

Ratios of 0.25-0.35 are considered quite large, but for lower pressure and velocity applications, the

ratios should be much smaller.

5.2. Multilayer reservoir problem. One may not always encounter constant permeability within

the subsurface. Some layers within the oil reservoir may consist of coarse sands while others may

consist of less permeable material. This numerical experiment shall study the effect varying per-

meability regions has on the pressure contours, flow rates, and local mass balance errors. Consider

the domain depicted in Figure 19 with the same boundary conditions as that in Figure 11. Regions

with higher permeability have larger velocities as depicted in Figure 20. The parameters used

for this problem are listed in Table 9, and the pressure contours for LS and VMS formalisms are

depicted in Figures 21 and 22 respectively.

Table 9. User-defined inputs for the layered reservoir problem

Parameter Value

βB 0.005

βF 0.01

k varies

µ0 1

ρ 1

ϑ 1

b(x)

0;−1

Nele 3200

penh 1000

31

Table 10. Layered reservoir problem: flow rates for LS and VMS formalism at penh

= 1000

Darcy models: D MB F MBF

LS 1038 210 133 75

VMS 1025 204 137 77

Results show that the layers with higher permeability contain higher pressures and that steep

gradients occur at the interfaces between the layers. The LS formalism predicts higher pressures

and larger flow rates for the Darcy and modified Barus models as seen from Table 10. Like with

the previous oil reservoir problem, the VMS formalism predicts higher flow rates for the Darcy-

Forchheimer model. The ratio of local mass balance errors and total predicted fluxes are depicted

in Figures 23 and 24. While the VMS formalisms still have slightly higher errors, the overall error

ratios for this problem are smaller despite having larger flow rates.

5.3. Flow in a porous media with staggered impervious zones. Consider flow through a

region with staggered impervious zones in Figure 25. In any heterogeneous flow through porous

media applications, one may encounter domains where oil must flow through a complex domain

with many impervious regions. The qualitative velocity vector field in Figure 26 indicates that

higher flows occur around the sharp bends. The pressure contours are depicted in Figures 27 and

28. The same non-dimensionalized injection pressure has been prescribed for this problem (see

Table 11 for key parameters used in this problem), and it can still be seen that the different Darcy

models make an impact on the qualitative nature of the pressure contours. Again, the LS formalism

Table 11. User-defined inputs for the staggered impervious zones problem

Parameter Value

βB 0.005

βF 0.01

k 1

µ0 1

ρ 1

ϑ 1

b 0

Element type Q9

Nele 1696

penh 1000

32

Table 12. Staggered impervious zones problem: flow rates for LS and VMS for-

malism at penh = 1000

Darcy models: D MB F MBF

LS 150.6 31.9 50.4 24.3

VMS 131.1 26.5 47.9 20.3

yields higher pressures throughout the domain and predicts larger flow rates as seen in Table 12.

Errors in the local mass balance tend to be greatest in regions with high velocities (i.e., the sharp

bends around the impervious layers). Local mass balancing errors are shown in Figures 29 and 30.

6. CONCLUDING REMARKS

The work in this thesis proposes a modification to the standard Darcy model that takes into

account both the dependence of the viscosity on the pressure and the inertial effects, which have

been observed in many physical experiments. The current models in the literature consider either

of the effects but not both. The proposed model will be particularly important for predictive

simulations of applications involving high pressures and high velocities (e.g., enhanced oil recovery).

This modification has been referred to as the modified Darcy-Forchheimer model. It has been shown

numerically that the results obtained by taking into account the dependence of drag coefficient on

the pressure and on the velocity are both qualitatively and quantitatively different from that the

results obtained using the standard Darcy model, Darcy-Forchheimer equation (which neglects the

dependence of drag coefficient and viscosity on the pressure) or modified Darcy model [3,20] (which

neglects the dependence of the drag coefficient on the velocity).

This thesis has also developed stable mixed finite element formulations for the resulting governing

equations using two different approaches: VMS formalism and LS formalism. Using numerical

experiments, we have compared their merits and demerits.

The LS formulation has more terms to evaluate than the VMS formulation, and hence the LS

formulation is slightly more computationally expensive than the VMS formulation. However, it

should be emphasized that this is not significant in a parallel setting as element-level calculations

are embarrassingly parallel. It is also observed that the LS formulation with p-refinement produces

accurate results. Another point that is worth mentioning is that the VMS formalism weakly

prescribes pressure boundary conditions, and it has been shown that minor oscillations occur when

meshes are not adequately refined. The error in element-wise / local mass balance for various Darcy-

type models is also quantified, and the error becomes significant when there are large pressures and

velocities.33

There are several ways one can extend the research work presented in this thesis. On the

modeling front, a good but difficult research problem is to develop mathematical models that

couple deformation and damage of the porous solid with the flow aspects and reactive transport

across several spatial and temporal scales. The following are some possible future works on the

numerical front :

(a) Develop mixed finite element formulations with better local mass balance property under equal-

order interpolation for the pressure and the velocity.

(b) Develop multi-scale models by coupling continuum / macro-scale flow models with meso-scale

models (e.g., lattice Boltzmann models). The advantage is that meso-scale models can easily

handle complex pore structure, which may not computationally feasible if one uses only a

macro-scale model.

(c) Another important but difficult problem is to develop numerical upscaling techniques for het-

erogeneous porous media. In layman terms, numerical upscaling captures fine-scale features on

coarse computational grids.

(d) Develop stable and accurate coupling algorithms for coupling flow, deformation and transport

aspects.

On the computer implementation front, a possible work is to implement the mixed formulations

taking the advantage of GPU processors, and implementing on heterogeneous parallel computing

environment.

ACKNOWLEDGMENTS

The authors acknowledges the financial support from the Department of Energy. The opinions

expressed in this paper are those of the authors and do not necessarily reflect that of the sponsors.

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37

Production wellInjection well Injection well

H=1

p(x)=penh

p(x)=p e

nh

p(x) = patmvn(x) = 0 vn(x) = 0

Γp

Γp Γp

Γv

Γv Γv

W = 0.1

W

vn(x) = 0

L = 2

L = 2

H=1

Figure 11. Oil reservoir problem. Top figure is the pictorial description of en-

hanced oil recovery, and the bottom figure is the idealized computational domain

with appropriate boundary conditions.

0 1 2

0

0.2

0.4

0.6

0.8

1

x

y

Figure 12. Oil reservoir problem: qualitative velocity vector field

38

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 13. Oil reservoir problem: pressure contours using LS formalism

39

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 14. Oil reservoir problem: pressure contours using VMS formalism

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−100

0

100

200

300

400

500

600

700

800

900

1000

x

Pre

ssur

e

DMBFMBF

(a) LS formalism

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−100

0

100

200

300

400

500

600

700

800

900

1000

x

Pre

ssur

e

DMBFMBF

(b) VMS formalism

Figure 15. Oil reservoir problem: comparison of pressure profiles at y = 1

41

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

Pressure

Flo

w r

ate

DMBFMBF

(a) LS formalism

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

Pressure

Flo

w r

ate

DMBFMBF

(b) VMS formalism

Figure 16. Oil reservoir problem: comparison of injection pressures vs. flow rates

42

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 17. Oil reservoir problem: ratios of local mass balance error over total

predicted flux using LS formalism

43

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 18. Oil reservoir problem: ratios of local mass balance error over total

predicted flux using VMS formalism

44

H=1

L = 2

k = 1

k = 10

k = 0.5

k = 10

k = 1

Figure 19. Layered reservoir problem: A pictorial description.

0 1 2

0

0.2

0.4

0.6

0.8

1

x

y

Figure 20. Layered reservoir problem: qualitative velocity vector field

45

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 21. Layered reservoir problem: pressure contours using LS formalism

46

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 22. Layered reservoir problem: pressure contours using VMS formalism

47

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 23. Layered reservoir problem: ratios of local mass balance error over total

predicted flux using LS formalism

48

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 24. Layered reservoir problem: ratios of local mass balance error over total

predicted flux using VMS formalism

49

p(x)=

p1 p

(x)=

p2

0.3 0.1 0.4 0.1 0.3

0.3

0.4

0.3

Figure 25. Staggered impervious zones problem: A pictorial description.

0 0.6 1.2

0

0.5

1

x

y

Figure 26. Staggered impervious zones problem: qualitative velocity vector field

50

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 27. Staggered impervious zones problem: pressure contours using LS formalism

51

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 28. Staggered impervious zones problem: pressure contours using VMS formalism

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(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 29. Staggered impervious zones problem: ratios of local mass balance error

over total predicted flux using LS formalism

53

(a) Darcy model (b) Modified Barus

(c) Darcy-Forchheimer (d) Modified Darcy-Forchheimer Barus

Figure 30. Staggered impervious zones problem: ratios of local mass balance error

over total predicted flux using VMS formalism

54